Hi Andrew,

I'm certainly no gear expert, I just have read and studied making gears for quite awhile so I'll try to explain the helical gear conundrum as best I can.

First of all you are correct in stating that the tooth and space of a spur gear are equal at the pitch circle. So let's take that a little further. Gear teeth are formed by the involute curve, that is no matter what number of teeth are on the mating pair they have a smooth rolling transition as they mesh, from rack to 13 teeth. There are 8 cutters per pitch to create the proper involute curve for each tooth count. Now I have never tried it but I'm assuming that from your statement of equal tooth width and spacing if you cut 2 gears with the same tooth count but used different numbered cutters on them the width and space would be the same at the PD. The difference is they wouldn't roll together smoothly because the mating tooth profiles is incorrect. I hope that makes sense.

Now onto helical gears. Here again going back to what you stated, tooth width/spacing being the same, helical gears are the same as spur gears in that respect. Now I will try and explain what is happening. Let's say that you set up to cut a helical path on a shaft but for the sake of explanation you use a woodruff key cutter to cut the groove. Given the diameter of the cutting head and the helical path it is traveling the sides of the groove will be splayed out. (wider at the top) The only way to cut an exact profile would be to use an end mill that was ground to the tooth profile and the cutting action was directly at the point of contact. This is impractical so involute cutters are used.

Now we know that each involute cutter in the set has a different profile and that cutting a helical path on a shaft with a large diameter cutter will produce a splayed slot we choose a cutter from the set that compensates for the splayed cut. There is a formula for finding NTCS (number of teeth for cutter selection) which is NTCS=NT x cube of the secant of the helix angle. There are charts that simplify that math.

The bottom line is the tooth and spacing for a given pitch are equal at the PD. it's just that you have to use the proper cutter to get the correct tooth profile for a given tooth count.

I have two free books that I downloaded years ago, one is a U.S. Navy machinists book the other is A Treatise on Milling by the Cincinnati machine company. Each of them cover gear cutting to the point that your head starts swimming but the basics are there and for the hobbyists it helps to explain how to cut gears more effectively.

gbritnell